Geostatistical analysis of functional data

Resumo

Functional data analysis concerns with statistical modeling of random variables taking values in a space of functions (functional variables), Several standard statistical techniques such as regression, ANOVA or principal components, among others, have been considered from a functional point of view. In general, these methodologies are focused on independent and identically distributed functional variables. However, in several disciplines of applied sciences there exists an increasing interest in modeling correlated functional data. In particular in most of them the modeling of spatially correlated functional data is of interest. This is the topic treated here. Specifically this work concerns with spatial prediction of curves when we dispose of a sample of curves collected at sites of a region with spatial continuity. Four methods for doing spatial prediction of functional data are developed. Initially, we propose a predictor having the same form as the classical kriging predictor, but considering curves instead of one-dimensional data. The other predictors arise from adaptations of functional linear models for functional response to the case of spatially correlated functional data. One the one hand, we define a predictor which is a combination of kriging and the functional linear point-wise (concurrent) model. On the other hand, we use the functional linear total model for extending two classical multivariable geostatistical methods to the functional context. The first predictor is defined in terms of scalar parameters. In the remaining cases the predictors involves functional parameters. We adapt an optimization criterion used in multivariable spatial prediction in order to estimate scalar and functional parameters involved in the predictors proposed. In all cases a non-parametric approach based on expansion in terms of basis functions is used for getting curves from discrete data. The number of basis functions is chosen by cross-validation. The methodologies proposed are illustrated by analyzing three real data sets corresponding to curves of penetration resistance and temperature which are functions of depth and time, respectively.